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I am trying to prove that $1-\xi_p$ is irreducible in $\mathbb{Z}[\xi_p]$, where $\xi_p$ is a primitive $p$-root of unity.

I am not sure how to do this. I have calculated the norm of $1-\xi_p$ and I know it is equal to the prime $p$. But I am not sure if I can conclude from this. I know that if an element of $\mathbb{Z}[\xi_p]$ is a unit then its norm is $+1$ or $-1$. But I am not sure whether having norm $+1$ or $-1$ implies that an element is a unit, which is what I would need to be able to conclude that $1-\xi_p$ is irreducible from the fact that it has norm a prime.

Any clarification will be very useful. Thanks in advance.

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    $\begingroup$ "But I am not sure whether having norm $+1$ or $-1$ implies that an element is a unit," - yes, it is true. See this duplicate. Or this one. $\endgroup$ Dec 29, 2022 at 18:55

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Yes: For a number field $k$, any $x\in {\cal O}_k$ with $N_{k/\mathbb{Q}}(x) = \pm 1$ is invertible. For if $f(x) = x^n + \cdots + a_1 x + a_0$ is the minimal polynomial of $x$, then $x(x^{n-1} + \cdots + a_1) = -a_0$ with $-a_0 = \pm N_{k/\mathbb{Q}}(x)\in {\cal O}_k^\times$.

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